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Gauss' Formula

German mathematician C.F. Gauss is often credited with discovering a formula for calculating the sum of a series when he was a young child. The story is likely apocryphal (a legend), but it has been passed down since Gauss lived in the 1700’s. According to the story, Gauss’s teacher wanted to occupy students by having them add up large sets of numbers. When Gauss was asked to add up the first 100 integers, he found the sum very quickly, by pairing the numbers:

All of the numbers in the sum could be paired to make groups of 101. There are one hundred numbers being added, so there are such fifty pairs. Therefore the sum is 50(101) = 5050.

The method Gauss used to solve this problem is the basis for a formula that allows us to add together the first
n
positive integers:

Using a graphing calculator

To generate either a sequence or a series, you can use a graphing calculator. The TI-83/84 series gives you several options. You can, for example, work in sequence mode, which allows you to define a sequence and find terms. If you have an explicit formula for a sequence, you can keep your calculator in function mode. For example, consider the sum
. It would be time consuming to write out the first 9 squares. The calculator is faster. To generate the 9 terms, press <TI font_2nd> [LIST], then select
OPS
, then option 5,
seq
(. This takes you back to the main screen. You should see
seq
(. After this, enter
x
^2,
x
, 1, 9, 1). (The
x
tells the calculator that
x
is the input. The 1 and the 9 tell it the limits of the sum. The second 1 tells the calculator to go up in increments of 1.)

Press <TI font_ENTER>, and you should see the list of squares. Scroll to the right to see all of them. The scrolling will end when you reach 81.

If you want to find the sum of the terms, first store the sequence in a list (see screen below), then <TI font_2nd> [LIST], then select
MATH
, then option 5,
sum(.
Then
enter the name of the list and press <TI font_)><TI font_ENTER>. You should get 285.

Example A

Expand the sigma and find the sum.

Solution

Example B

Expand the sigma and find the sum by adding the terms

Why is Gauss' formula not recommended for this question?

Solution

There are actually a couple of reasons not to use Gauss formula here, but the biggest is that the formula assumes you are adding all of the integers from 0 to the last number in the series. In the question, you are asked to only sum from the 3rd to 6th term.

Example C

If the sum of the first
n
integers is 210, what is
n
?

Solution

Using Gauss' formula:

Multiply both sides by 2

Distribute

Complete the square

Factor

Square root both sides

Concept question wrap-up

The series sum formula
is designed for integers, so let's use it to solve for the number of
dimes
brought in (since that is the unit each term reduces by) and then convert to dollars:

The store will bring in $183.00, which will probably
not
cover the costs of the day. However, they will certainly get a lot of people through the door to try out the ice cream!

Vocabulary

If a series has a limit, and the limit exists, the series
converges
.

If a series does not have a limit, or the limit is infinity, then the series
diverges
.

The
index
of the sum is the variable in the sum.

The
limits
of a summation are the starting and ending points of the sum.

Sigma
is the Greek letter used to represent a sum.

The
summand
of a sigma is the expression being summed.

Guided Practice

Questions

1) Express the sum using sigma notation: 1 + 3 + 9 + 27 + ...

2) Find the sum of

3) Use Gauss’s formula to find the sum of the first 200 positive integers.

4) Expand the sigma and find the sum.

Solutions

1)
or

2) Let's look at both ways of solving this one:

a) We could plug in all the numbers between 0 and 12 to get:
... and then add them together to get the sum

b) Using a formula:

A slightly modified version of Gauss' formula looks this:
where
k
is the number of terms in the series
plus one
, and
a
0
and
a
n
are the first and last terms in the series

3) Using Gauss' formula:

n
= 200

Sum = 20,100

4) For

Practice

Calculate the sums of the given series, you may use addition of individual terms, or a series sum formula. You may use a graphing tool for any 3 of them. Try to use each method at least once.

Consider the sums
and
.

What is the product of

What is the sum of
?

Look closer at the last two problems, what does this tell you about rules for working with sums?

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Description

Calculating the sum of the terms in a series.

Learning Objectives

Here you will learn of a formula that makes it easier to find the sum of the terms in a series, and will further explore the use of summation notation and number series.